What Is a Vanishing Point? Where Parallel Lines Meet on a Flat Surface

Perspective drawing treats the picture surface as a window and constructs what a viewer at a fixed point would see through it. Any line in the world projects onto that surface as a line, and a set of parallel lines projects to a set that converges on one point, since a viewer looking along the direction those lines run sees them apparently meeting at the horizon. That convergence point depends only on the direction of the lines, not on where they are, so every set of parallels running in a given direction shares one vanishing point. Horizontal lines vanish on the horizon, which sits at the viewer's eye level, and lines running in other directions vanish elsewhere. Lines parallel to the picture surface do not converge at all, since their vanishing point is infinitely far off to the side.

Practical drawing describes arrangements by how many vanishing points are used:

• •One-point perspective, where the subject is square to the viewer and only lines running away converge, used for corridors and interiors
• •Two-point, where the subject is at an angle so two sets of horizontal lines converge to points on the horizon
• •Three-point, adding a vertical vanishing point above or below for looking up or down at tall subjects
• •Curvilinear systems, which handle very wide angles where straight-line perspective distorts badly
• •Aerial or atmospheric perspective, which is about colour and contrast rather than geometry and works alongside it
• •Multiple vanishing points for objects at different angles within one picture, which is the ordinary case rather than an exception

Linear perspective was demonstrated in Florence in the early fifteenth century, with Brunelleschi conducting a public demonstration involving a painted panel and a mirror that allowed a viewer to compare the painting with the building it depicted. Alberti set out the method in writing shortly afterwards, giving painters a construction they could follow. The system spread rapidly and reshaped European painting within decades, and treatises elaborating it continued for centuries. It was not the first attempt at depicting depth, since earlier and non-European traditions used various conventions including overlapping, diminishing size and oblique projection that indicates depth without convergence, and describing those as failures misunderstands them, since a tradition may not be attempting the particular illusion that this method produces.

Drawing in perspective is a procedure and the steps are mechanical once the setup is chosen. The horizon is drawn at the intended eye level, which decides whether the viewer is above or below the subject and is the first and most consequential decision. Vanishing points are placed on it for each set of horizontal directions in the scene. Lines are then drawn from each point to establish the receding edges. Spacing objects correctly along a receding surface is the part that catches people out, since equal intervals in the world are unequal on the page, and a standard construction using the diagonals of a square solves it by generating each successive interval from the last. Measuring points allow real dimensions to be transferred. Photographic lenses produce the same geometry automatically, which is why perspective can be checked by comparison with a photograph taken from the same position.

The method rests on assumptions that are worth stating, since they are why it is a convention rather than a neutral record. It assumes a single eye at a fixed point, which nobody has, and a picture viewed from exactly that point, which almost nobody does, so a perspective picture is technically correct only from one position and is accepted from others because the visual system tolerates it. It assumes a flat picture surface. It produces conspicuous distortion at wide angles, with spheres near the edges of a wide view appearing stretched, which is geometrically correct and looks wrong. And it privileges one kind of spatial information over others, which is why artists have abandoned it deliberately, with several modern movements rejecting it precisely to assert that a picture is a flat object rather than a window.

Parallel lines project to lines converging on one point determined by their direction alone, with horizontal sets meeting on the horizon at the viewer's eye level. One, two and three-point arrangements describe how many are in use. The method was demonstrated in fifteenth-century Florence and assumes a single fixed eye, which is why the result is exactly correct from only one position.